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  Textbook on group theory to be able to start QFT

+ 3 like - 0 dislike

I am very enthusiastic about learning QFT. How much group theory would I need to master? Please could you recommend me a textbook on group theory, which would help me to start QFT?

This post imported from StackExchange Physics at 2014-03-24 04:54 (UCT), posted by SE-user ramanujan_dirac

asked Apr 8, 2012 in Resources and References by ramanujan_dirac (235 points) [ revision history ]
recategorized Apr 24, 2014 by dimension10
Cross-listed from http://physics.stackexchange.com/q/23387/2451

This post has been migrated from (A51.SE)
There is a list here. I would also recommend Differential Geometry and Lie Groups for Physicists and Group Theory: A Physicist's Survey

This post imported from StackExchange Physics at 2014-03-24 04:54 (UCT), posted by SE-user Vijay Murthy
Cross-listed to theoreticalphysics.stackexchange.com/q/1105/189

This post imported from StackExchange Physics at 2014-03-24 04:54 (UCT), posted by SE-user Qmechanic

There is now a (relatively) new "nutshell" from Antony Zee for group theory


5 Answers

+ 3 like - 0 dislike

If you are interested more in physics than maths then I recommend: http://www.amazon.com/Lie-Algebras-Particle-Physics-Frontiers/dp/0738202339 as a start.

This post has been migrated from (A51.SE)
answered Apr 8, 2012 by Kyle (335 points) [ no revision ]
+ 3 like - 0 dislike

Georgi's book on Lie Groups is enough, but most of the group theory is explained in the physics texts. It is nice to learn group theory, but the mathematician's theory is more concerned with characters and root lattices, which are nice, but not essential in most of the bread-and-butter applications. The ALE classification is important in mathematical physics, but I think it is covered properly in the physics literature.

You don't need anything too special--- just the rudiments of Lie groups (it doesn't hurt to know group theory, though, it is just not essential). You can learn everything on your own from the QFT source and thinking it out--- there SU(2)/SU(3) cases are not too bad, and these are about as big as it gets. SU(5) and E8 require more sophistication, but are best covered in GUT papers and Green/Schwarz/Witten (for a great introduction to E8)

The modern algebra you probably want to learn is not group theory, but homological algebra, category theory, and Hopf algebra. These are covered well by Lang's algebra book, which is a graduate school staple in mathematics. It doesn't hurt to know everything in Lang--- it's well written, as everything by Lang--- although a little philosophically annoying for me, because it is so conservative in its set-theoretic appratus.

This post imported from StackExchange Physics at 2014-03-24 04:54 (UCT), posted by SE-user Ron Maimon
answered Apr 8, 2012 by Ron Maimon (7,730 points) [ no revision ]
+ 2 like - 0 dislike

Usually, the material is covered self-consistently in the textbooks on QFT themselves. One can look at the beginning of the first volume of Quantum Theory of Fields by Weinberg, for example, where even projective representations are discussed. Non-abelian symmetries and supergroups are covered in the second and third volumes respectively. Quite a comprehensive introduction to all of these topics can also be found in Fields by Siegel (http://insti.physics.sunysb.edu/~siegel/Fields4.pdf). Here a conformal group is also discussed in details. Very accessible and geometrical introduction is given in Rubakov's "Classical Theory of Fields".

More advanced topics, such as weights, roots, Dynkin diagrams, etc. could be found in textbooks on String Theory (Johnson "D-branes", Polchinski "String Theory", many others).

answered Feb 11, 2017 by Andrey Feldman (904 points) [ revision history ]
edited Feb 13, 2017 by Andrey Feldman
+ 1 like - 0 dislike

Actually to start learning the basics of QFT you do not need so much group theory [a different thing is if you want to go to the details]. Some of the introductory books in QFT have at the beginning a section about Lorentz and Poincaré groups, scalar, tensor and spinor representation etc. This is the case, for example in Maggiore, A Modern Introduction to Quantum Field Theory.

If you want group theory for physics for its own sake [which I find useful and beautiful] you may want to learn Lie groups and representations. Start with the link given by Vijay Murthy. There are also very good courses in the internet, I can recommend you some if you want.

This post imported from StackExchange Physics at 2014-03-24 04:54 (UCT), posted by SE-user DaniH
answered Apr 8, 2012 by DaniH (60 points) [ no revision ]
+ 0 like - 0 dislike

Contemporary Abstract Algebra by Joseph Gallian is a good introduction to group theory.

This post imported from StackExchange Physics at 2014-03-24 04:54 (UCT), posted by SE-user Vafa Khalighi
answered Apr 8, 2012 by Vafa Khalighi (0 points) [ no revision ]

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